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Consider the complex function f(z) = sin(z)/z.
What kind of isolated singularity does it have at 0?

removable
====================
You can prove your answer by referring to the definition only.
This is also an opportunity to test the unproven theorem. (MRW
2001.02.27)
THEOREM: G c |C; f : G -> |C; b:-|C
thesis: If function f has an isolated singularity at poin

Consider the complex function f(z) = ( cos(z) - 1 ) / z.
What kind of isolated singularity does it have at 0?

removable
====================
You can prove your answer by referring to the definition only.
This is an opportunity to test the unproven theorem. (MRW
2001.02.04)
THEOREM: G c |C; f : G -> |C; b:-|C
thesis: If function f has an isolated singularity at poin

lim(x->0,y->0) (x-sin(y))*y^2 / (y^2 + sin(x-y)*sin(x-y) ) = ???

= 0
hint: y^2 <= y^2 + sin(x-y)*sin(x-y)

lim(x->0) x*sin(x) / ( x^2 + sin(x)*sin(x) ) = ???

= 1/2
hint: use the squeeze theorem

Consider the complex function f(z) = (z+1)/z.
Does it have a removable isolated singularity at 0 ?
Justify your answer in two different ways.